Design: Blocked RCT, with 2 levels, and randomization done at level 1 (individual level).

Models: Constant treatment effects, fixed treatment effects, and random treatment effects.

Power Validation

Default parameters:

  • M = 3
  • J = 20
  • rho: \(\rho\) = 0.5
  • ATE ES = 0.125, 0.125, 0.125
  • R2.1: \(R^2_1\) = 0, 0, 0

Parameters by model type:

  • Omega: \(\omega_2 = 0\) for constant effects, \(\omega_2\) = for fixed and random
  • ICC: \(\text{ICC}_2 = 0\) for constant and fixed effects, \(\text{ICC}_2\) = 0.2, 0.2, 0.2 for random effects

Assumptions:

  • Two-level design: \(\text{ICC}_3 = 0\), \(\omega_3 = 0\), \(K = 1\)

Varying school size

\(\bar{n}\) = 50

\(\bar{n}\) = 50

\(\bar{n}\) = 50

Varying R2

\(R^2_1 =\) 0.6, 0.6, 0.6

\(R^2_2 =\) 0.6, 0.6, 0.6 (This should not affect power!)

Varying rho

\(\rho =\) 0.2

\(\rho =\) 0.8

Varying true positives

ATE (ES) = 0.125, 0, 0

Varying ICC

\(\text{ICC}_2\) = 0.8, 0.8, 0.8

MDES validation

## 
## 
## +------------+---------------+-------------+---------------+
## |    MTP     | Adjusted MDES | Indiv Power | Targeted MDES |
## +============+===============+=============+===============+
## |    rawp    |     0.126     |    0.509    |     0.125     |
## +------------+---------------+-------------+---------------+
## | Bonferroni |     0.124     |    0.334    |     0.125     |
## +------------+---------------+-------------+---------------+
## |     BH     |     0.124     |    0.418    |     0.125     |
## +------------+---------------+-------------+---------------+
## |    Holm    |     0.127     |    0.401    |     0.125     |
## +------------+---------------+-------------+---------------+
## 
## Table: blocked_i1_2c
## 
## 
## +------------+---------------+-------------+---------------+
## |    MTP     | Adjusted MDES | Indiv Power | Targeted MDES |
## +============+===============+=============+===============+
## |    rawp    |     0.124     |     0.5     |     0.125     |
## +------------+---------------+-------------+---------------+
## | Bonferroni |     0.124     |    0.331    |     0.125     |
## +------------+---------------+-------------+---------------+
## |     BH     |     0.125     |    0.429    |     0.125     |
## +------------+---------------+-------------+---------------+
## |    Holm    |     0.124     |    0.384    |     0.125     |
## +------------+---------------+-------------+---------------+
## 
## Table: blocked_i1_2f
## 
## 
## +------------+---------------+-------------+---------------+
## |    MTP     | Adjusted MDES | Indiv Power | Targeted MDES |
## +============+===============+=============+===============+
## |    rawp    |     0.124     |    0.428    |     0.125     |
## +------------+---------------+-------------+---------------+
## | Bonferroni |     0.125     |    0.243    |     0.125     |
## +------------+---------------+-------------+---------------+
## |     BH     |     0.126     |    0.346    |     0.125     |
## +------------+---------------+-------------+---------------+
## |    Holm    |     0.123     |    0.288    |     0.125     |
## +------------+---------------+-------------+---------------+
## 
## Table: blocked_i1_2r

Sample size validation

## 
## 
## +------+-------------+-------------+---------------+
## | MTP  | Sample Type | Sample Size | D1indiv power |
## +======+=============+=============+===============+
## |  BH  |      J      |     21      |     0.444     |
## +------+-------------+-------------+---------------+
## | Holm |      J      |     21      |     0.413     |
## +------+-------------+-------------+---------------+
## |  BH  |    nbar     |     50      |     0.426     |
## +------+-------------+-------------+---------------+
## | Holm |    nbar     |     52      |     0.409     |
## +------+-------------+-------------+---------------+
## 
## Table: blocked_i1_2c
## 
## 
## +------+-------------+-------------+---------------+
## | MTP  | Sample Type | Sample Size | D1indiv power |
## +======+=============+=============+===============+
## |  BH  |      J      |     20      |     0.415     |
## +------+-------------+-------------+---------------+
## | Holm |      J      |     20      |     0.394     |
## +------+-------------+-------------+---------------+
## |  BH  |    nbar     |     50      |     0.423     |
## +------+-------------+-------------+---------------+
## | Holm |    nbar     |     52      |     0.402     |
## +------+-------------+-------------+---------------+
## 
## Table: blocked_i1_2f
## 
## 
## +------+-------------+-------------+---------------+
## | MTP  | Sample Type | Sample Size | D1indiv power |
## +======+=============+=============+===============+
## |  BH  |      J      |     20      |     0.328     |
## +------+-------------+-------------+---------------+
## | Holm |      J      |     20      |     0.288     |
## +------+-------------+-------------+---------------+
## |  BH  |    nbar     |     50      |     0.339     |
## +------+-------------+-------------+---------------+
## | Holm |    nbar     |     49      |     0.28      |
## +------+-------------+-------------+---------------+
## 
## Table: blocked_i1_2r